Fokker–Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces

We consider a Kolmogorov operator L0 in a Hilbert space H, related to a stochastic PDE with a timedependent singular quasi-dissipative drift F = F(t, ·) :H → H, defined on a suitable space of regular functions. We show that L0 is essentially m-dissipative in the space Lp([0,T] × H; ν), p 1, where ν(dt,dx) = νt (dx) dt and the family (νt )t∈[0,T ] is a solution of the Fokker–Planck equation given by L0. As a consequence, the closure of L0 generates a Markov C0-semigroup. We also prove uniqueness of solutions to the Fokker–Planck equation for singular drifts F. Applications to reaction–diffusion equations with time-dependent reaction term are presented. This result is a generalization of the finite-dimensional case considered in [V. Bogachev, G. Da Prato, M. Röckner, Existence of solutions to weak parabolic equations for measures, Proc. London Math. Soc. (3) 88 (2004) 753–774], [V. Bogachev, G. Da Prato, M. Röckner, On parabolic equations for measures, Comm. Partial Differential Equations 33 (3) (2008) 397–418], and [V. Bogachev, G. Da Prato, M. Röckner, W. Stannat, Uniqueness of solutions to weak parabolic equations for measures, Bull. London Math. Soc. 39 (2007) 631–640] to infinite dimensions. © 2008 Elsevier Inc. All rights reserved.